Single-ensemble multilevel Monte Carlo for discrete ensemble Kalman methods
Arne Bouillon, Toon Ingelaere, Giovanni Samaey
TL;DR
The paper addresses the high computational cost of ensemble Kalman methods for state and parameter estimation when the forward model is expensive. It introduces a single-ensemble MLMC framework that uses a hierarchy of forward-model approximations $\{\mathcal{G}_\ell\}$ and performs MLMC at each time step to estimate interaction terms, extending previous single-ensemble MLMC work for EnKF to a broader class of EnKF methods. Under a set of mild assumptions on the forward-model approximations, interaction maps, and estimators, the authors derive convergence rates showing that the multilevel method achieves optimal cost-to-accuracy scaling relative to standard Monte Carlo, with the rate depending on the bias decay exponent $\beta$ and the cost growth exponent $\gamma$. Numerical scaling experiments on Ornstein–Uhlenbeck state estimation and Darcy-flow Bayesian inversion corroborate the predicted rates and demonstrate practical improvements, while the authors discuss open questions for non-asymptotic bounds and further extensions to multiple ensembles.
Abstract
Ensemble Kalman methods solve problems in domains such as filtering and inverse problems with interacting particles that evolve over time. For computationally expensive problems, the cost of attaining a high accuracy quickly becomes prohibitive. We exploit a hierarchy of approximations to the underlying forward model and apply multilevel Monte Carlo (MLMC) techniques, improving the asymptotic cost-to-error relation. More specifically, we use MLMC at each time step to estimate the interaction term in a single, globally-coupled ensemble. This technique was proposed by Hoel et al. for the ensemble Kalman filter; our goal is to study its applicability to a broader family of ensemble Kalman methods.